Power

Average power: \(P_{avg}=\frac{W}{\Delta t}\). Instantaneous power: \(P=\frac{dW}{dt}\).

• Units: watts (J/s)
• When average is enough
• Meaning of “instantaneous”

For a force doing work on an object: \(P = \vec F \cdot \vec v\). Only the component along velocity contributes.

• Dot product meaning
• Perpendicular force → zero power
• Sign: delivering vs removing energy

Apply power to engines, elevators, cycling, and braking. Relate power to energy change rates.

• Lifting at constant speed
• Accelerating systems
• Power and performance limits

Efficiency compares useful output power to input power: \(\eta = \frac{P_{out}}{P_{in}}\).

• Losses (heat, sound, friction)
• Why \(\eta < 1\)
• Interpreting specs

Solve power problems involving work rates, force–velocity, and efficiency.

• Average power computations
• Instantaneous \(P=\vec F\cdot\vec v\) drills
• Elevator/engine scenarios
• Efficiency word problems
• Concept checks: direction/sign

Work changes energy. Power measures how quickly that change happens.

• Work–energy theorem recap
• Conservation + losses
• Power as rate